Progress and Problems in Real-Space Renormalization

James Jeffrey Binney, N J Dowrick, Andrew J. Fisher +1 more

Abstract In the previous chapter we saw that it is possible to make a great deal of progress in the study of particular models by real-space renormalization methods. These methods are very elegant and can explain why critical exponents arise and how they are related to each other. However, they are also very specific: it requires considerable ingenuity to invent an accurate (and tractable) real-space scaling transformation for any given model, and even when one has succeeded it is not normally very helpful in tackling the next problem.

Theodore W. Burkhardt, J. M. J. van Leeuwen

Patricio Cordero A

Ulf Sannemo, K. A. Chao

We have performed a real-space renormalization calculation to investigate the low-temperature properties of 54 randomly located quantum spins (s=(1/2) with Ruderman-Kittel-Kasuya-Yosida interactions. The computed spin-glass order parameter, dc magnetic susceptibility, magnetic specific heat, internal field distribution, and spatial correlation of eigenstates suggest a progressive freezing of spins with decreasing temperature.

Jeffrey J. Prentis

This dissertation involves the renormalization group theory and its application to two systems exhibiting renormalization symmetry. Both the real space and the field theoretic formulations of the renormalization group theory are discussed. A continuum of real space renormalization transformations is used to study the critical behavior of the two-dimensional Ising model. We find that there exists an optimal transformation that produces sharply more accurate critical properties that the others. We establish a set of conditions that characterize the optimal transformation. These conditions suggest criteria that could enable one to control a given real space renormalization approximation. Field theory renormalization is used to exploit the renormalization symmetry in the excluded volume problem. We develop a self-contained formalism that incorporates the concepts and methods of field theory renormalization directly into the language of the excluded volume problem. This formalism avoids the Lagrangian model isomorphism, thereby providing a simpler and more natural framework for underst and ing the excluded volume problem from first principles. This formalism is used to investigate the spatial correlations in a large self-repelling ring polymer. In particular, we find that the ratio of the radius of gyration of a ring polymer to that of a linear polymer having the same number of monomers is a universal quantity whose value we calculate to first order in (epsilon) (TBOND) 4 - (dimension of space). In addition, we study the mutual excluded volume characterizing a system of two chains with different lengths. The renormalization of the polymer perturbation theory through two-loop order enables us to calculate the universal value of the interpenetration function to second order in (epsilon). Our renormalization group results are in very good agreement with the results obtained from numerical and experimental studies of rings and chains in three dimensional space. We also discuss the connection between real space and field theory renormalization. We outline the basic conceptual ideas that underlie the general theory of renormalization and illustrate how each of these ideas is realized in the real space and the field theoretic formulations.

Manfred Salmhofer B